Parqueting-Reflection Principle and Boundary Value Problems in Some Circular Polygons [thesis]

Hanxing Lin, Universitätsbibliothek Der FU Berlin
2021
This dissertation is an investigation of the theory of the parqueting-reflection principle and its applications to basic boundary value problems in some circular polygons. The parqueting-reflection principle is applied to solve several boundary value problems for particular domains whose boundaries are composed of circular arcs. It provides heuristic ideas and procedures for constructing harmonic Green functions and harmonic Neumann functions, which play important roles in dealing with
more » ... and Neumann boundary value problems for the Poisson equation. The parqueting-reflection principle also contributes a method to solve Schwarz boundary value problems for the homogeneous and inhomogeneous Cauchy-Riemann equations. The parqueting-reflection principle has been verified to successfully solve these boundary value problems for many planar domains. However, this principle has not yet been well explained or rigorously justified in theory. This dissertation dedicates to building a fundamental theory for the parqueting-reflection principle and exploring new domains in which the principle can be applied. The main works of this dissertation are listed below. We first discuss circle reflections in the extended complex plane and employ some matrix techniques in dealing with circle reflections. These matrix tools bring some convenience for the discussions and the computations. Some results on consecutive circle reflections are also prepared for further discussions. We next introduce the definition of parqueting-reflection domains, in which the parqueting-reflection principle is supposed to be applicable. We prove that the parqueting-reflection principle succeeds in constructing the harmonic Green and Neumann functions for finite parqueting-reflection domains. We also obtain some properties of the normal derivatives of harmonic Green and Neumann functions on the boundary of the domains. We then fully overview basic boundary value problems in disks and half-planes and unify the harmonic Green and Neumann functions, the Sch [...]